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Computing canonical heights using arithmetic intersection theory


Author: Jan Steffen Müller
Journal: Math. Comp. 83 (2014), 311-336
MSC (2010): Primary 11G50; Secondary 11G10, 11G30, 14G40
DOI: https://doi.org/10.1090/S0025-5718-2013-02719-6
Published electronically: June 14, 2013
MathSciNet review: 3120591
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Abstract: For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height of a point on the Jacobian of a smooth projective curve can be computed using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.


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  • 1. W.W. Adams and P. Loustaunau, An introduction to Gröbner bases, American Mathematical Society, Providence, (1994). MR 1287608 (95g:13025)
  • 2. M. Artin, Lipman's proof of resolution of singularities for surfaces, in G. Cornell and J.H. Silverman (eds.), Arithmetic geometry, Springer-Verlag, New York-Heidelberg-Berlin (1986). MR 861980
  • 3. J. Balakrishnan, Coleman Integration for Hyperelliptic Curves: Algorithms and Applications, PhD thesis, MIT (2011).
  • 4. J. Balakrishnan and A. Besser, Local heights on hyperelliptic curves, Int. Math. Res. Notices. (2011), doi: 10.1093/imrn/rnr111.
  • 5. A. Bobenko, B. Deconinck, M. Heil, M. Schmies and M. van Hoeij, Computing Riemann Theta Functions, Math. Comp. 73, 1417-1442 (2004). MR 2047094 (2005c:65015)
  • 6. D. Cantor, Computing in the Jacobian of a Hyperelliptic Curve, Math. Comp. 48 (177), 95-101 (1987). MR 866101 (88f:11118)
  • 7. V. Cossart, U. Jannsen and S. Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, Preprint (2009). arXiv:math/0905.2191v2 [math.AG]
  • 8. D.A. Cox and S. Zucker, Intersection numbers of sections of elliptic surfaces, Invent. Math. 53, 1-44 (1969). MR 538682 (81i:14023)
  • 9. R.F. Coleman and B.H. Gross, $ p$-adic heights on curves, Algebraic Number Theory - in honor of K. Iwasawa, Advanced Studies in Pure Mathematics 17, 73-81 (1989). MR 1097610 (92d:11057)
  • 10. B. Deconinck and M. van Hoeij, Computing Riemann matrices of algebraic curves, Physica D 152-153, 28-46 (2001). MR 1837895 (2002j:30001)
  • 11. B. Deconinck and M. Patterson, Computing the Abel map, Physica D 237, 3214-3232 (2008). MR 2477016 (2010d:37139)
  • 12. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer-Verlag, New York (1995). MR 1322960 (97a:13001)
  • 13. G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. (2) 119, 387-424 (1984). MR 740897 (86e:14009)
  • 14. J.-C. Faugère, A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Algebra 139 (1), 61-88 (1999). MR 1700538 (2000c:13038)
  • 15. E.V. Flynn, N.P. Smart, Canonical heights on the Jacobians of curves of genus 2 and the infinite descent, Acta Arith. 79, 333-352 (1997). MR 1450916 (98f:11066)
  • 16. E.V. Flynn, F. Leprévost, E.F. Schaefer, W.A. Stein, M. Stoll and J.L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70, 1675-1697 (2001). MR 1836926 (2002d:11072)
  • 17. B. Gross, Local heights on curves, in G. Cornell and J.H. Silverman (eds.), Arithmetic geometry, Springer-Verlag, New York-Heidelberg-Berlin, (1986). MR 861983
  • 18. A. Hashemi and D. Lazard, Almost polynomial complexity for zero-dimensional Gröbner bases, in Proceedings of the 7th Asian Symposium on Computer Mathematics (ASCM'2005), Seoul, Korea, 16-21 (2005).
  • 19. F. Hess, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symbolic Comp. 33(4), 425-445 (2002). MR 1890579 (2003j:14032)
  • 20. D. Holmes, Computing Néron-Tate heights of points on hyperelliptic Jacobians, J. Number Theory (2012), doi:10.1016/j.jnt.2012.01.002
  • 21. D. Holmes, Néron-Tate heights on the Jacobians of high-genus hyperelliptic curves, PhD thesis, University of Warwick, 2012.
  • 22. P. Hriljac, Heights and Arakelov's intersection theory, Amer. J. Math. 107, 23-38 (1985). MR 778087 (86c:14024)
  • 23. S. Lang, Fundamentals of diophantine geometry, Springer-Verlag, New York (1983). MR 715605 (85j:11005)
  • 24. S. Lang, Introduction to Arakelov theory, Springer-Verlag, New York (1988). MR 969124 (89m:11059)
  • 25. Q. Liu, Algebraic Geometry and arithmetic curves, Oxford University Press, Oxford (2002). MR 1917232 (2003g:14001)
  • 26. MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp., 24, 235-265 (1997). (See also the Magma homepage at http://magma.maths.usyd.edu.au/magma/.) MR 1484478
  • 27. H. Matsumura, Commutative algebra, W.A. Benjamin, New York (1970). MR 0266911 (42:1813)
  • 28. B. Mazur and J. Tate, Canonical height pairings via biextensions, in Arithmetic and geometry, Vol. I, Progr. Math., 35, 195-237, Birkhäuser, Boston (1983). MR 717595 (85j:14081)
  • 29. J.S. Müller, Computing canonical heights on Jacobians, PhD thesis, Universität Bayreuth (2010).
  • 30. http://www.math.uni-hamburg.de/home/js.mueller/#code
  • 31. A. Néron, Quasi-fonctions et hauteurs sur les variétes abéliennes, Ann. of Math. 82, 249-331 (1965). MR 0179173 (31:3424)
  • 32. S. Pauli, Factoring polynomials over local fields, J. Symbolic Comput. 32, 533-547 (2001). MR 1858009 (2002h:13038)
  • 33. F. Pazuki, Minoration de la hauteur de Néron-Tate sur les variétés abéliennes: sur la conjecture de Lang et Silverman, PhD thesis, Université Bordeaux 1 (2008).
  • 34. B. Poonen and E. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488, 141-188 (1997). MR 1465369 (98k:11087)
  • 35. J. Romero-Valencia and A.G. Zamora, Explicit constructions for genus 3 Jacobians, Preprint (2009), arXiv:math/0904.4537v1[math.AG].
  • 36. P. Schneider, $ p$-adic height pairings I, Invent. Math. 69, 401-409 (1982). MR 679765 (84e:14034)
  • 37. J.H. Silverman, Computing heights on elliptic curves, Math. Comp. 51, 339-358 (1988). MR 942161 (89d:11049)
  • 38. M. Stoll, On the height constant for curves of genus two, II, Acta Arith. 104, 165-182 (2002). MR 1914251 (2003f:11093)
  • 39. M. Stoll, Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math 11, 367-380 (2008). MR 2465796 (2010b:11067)
  • 40. M. Stoll, Explicit Kummer varieties for hyperelliptic curves of genus three, to appear. See also http://www.mathe2.uni-bayreuth.de/stoll/talks/Luminy2012.pdf.
  • 41. http://www.math.lsu.edu/~wamelen/genus2.html
  • 42. M. Wagner, Über Korrespondenzen zwischen algebraischen Funktionenkörpern, PhD thesis, TU Berlin (2009).

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Additional Information

Jan Steffen Müller
Affiliation: Fachbereich Mathematik, Universität Hamburg
Email: jan.steffen.mueller@uni-hamburg.de

DOI: https://doi.org/10.1090/S0025-5718-2013-02719-6
Received by editor(s): June 29, 2011
Received by editor(s) in revised form: January 27, 2012, and March 5, 2012
Published electronically: June 14, 2013
Additional Notes: This work was supported by DFG-grant STO 299/5-1
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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